493 lines
20 KiB
Matlab
493 lines
20 KiB
Matlab
classdef ML_MLSE < handle
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% ALGORITHM DESCRIBED IN:
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% W. Lanneer and Y. Lefevre, “Machine Learning-Based Pre-Equalizers for
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% Maximum Likelihood Sequence Estimation in High-Speed PONs,”
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% in 2023 31st European Signal Processing Conference
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% Further ML Refs:
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% https://machinelearningmastery.com/cross-entropy-for-machine-learning/
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% https://docs.pytorch.org/docs/stable/generated/torch.nn.CrossEntropyLoss.html
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% The central idea is to overcome the (white-) noise assumption within the previously described
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% Viterbi algorithm, more precisely a closed-loop optimization is proposed that finds a suitable
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% filter-set to directly compute the branch metrics c_k (s,s^' ). These can directly be used to
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% carry out the conventional Viterbi algorithm. The system consists of S^L S=F linear FIR filters,
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% combined with one bias coefficient respectively. These filters take the received input samples to
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% compute the branch metrics estimates (c_k ) ̂(s,s^' ) according toThe central idea is to overcome
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% the (white-) noise assumption within the previously described Viterbi algorithm, more precisely
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% a closed-loop optimization is proposed that finds a suitable filter-set to directly compute the
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% branch metrics c_k (s,s^' ). These can directly be used to carry out the conventional Viterbi
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% algorithm. The system consists of S^L S=F linear FIR filters, combined with one bias coefficient
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% respectively. These filters take the received input samples to compute the branch metrics
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% estimates. Finally, the usual Viterbi is carried out...
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% Recommended Settings and some findings:
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% Requires many training epochs. According to ML people, 100,200 or
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% even up to 1000 epochs are normal for ML-convergence
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% The mu parameter _can_ be adaptive - using the cross entropy and when
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% analyzing the isolated training it looks very promisig. However, is
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% later use I found this is not as stable as a fixed learning rate.
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% mu = 0.1 worked good for me
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% Longer orders/ filter length are not always better. For me order=11
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% was good.
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% Delay factor (delta) is good when the order is also increased. With
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% order = 11, a delta of =4 shows good results
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properties
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sps % usually 2
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order
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e
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e_tr
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error
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len_tr
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mu_tr
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epochs_tr
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dd_mode % 1 or 0 to set DD-mode on or off
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mu_dd %weight update in dd mode
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epochs_dd
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adaptive_mu
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constellation
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L %viterbi memory length
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alpha
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DIR
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DIR_flip
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trellis_states
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traceback_depth
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% --- Added internal class variables used later ---
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S
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Nf
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delta
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nStates
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nFeasible
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combs
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first_sym
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last_sym
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valid
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valid_to_idx
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valid_from_idx
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w
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% --- New: fast state lookup ---
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true_to_state_idx
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state_dict % containers.Map: key(sequence)->state index
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key_fmt = '%.8g_'; % key format for sequence strings
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nSym % |constellation|
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ber = []
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ce = ones(1,1);
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end
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methods
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function obj = ML_MLSE(options)
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arguments(Input)
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options.sps = 2;
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options.order = 15;
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options.len_tr = 4096;
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options.mu_tr = 0;
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options.epochs_tr = 5;
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options.dd_mode = 1;
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options.mu_dd = 1e-5;
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options.epochs_dd = 5;
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options.adaptive_mu = 1;
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options.delta = 0;
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options.traceback_depth = 1024;
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options.L = 1
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end
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fn = fieldnames(options);
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for n = 1:numel(fn)
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obj.(fn{n}) = options.(fn{n});
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end
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obj.e = zeros(obj.order,1);
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obj.error = 0;
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end
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function [X,X_viterbi] = process(obj, X, D)
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% actual processing of the signal (steps 1. - 3.)
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% 1 normalize RMS
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X = X.normalize("mode","rms");
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% Use sorted constellation for deterministic mapping
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obj.constellation = sort(unique(D.signal),'ascend');
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obj.nSym = numel(obj.constellation);
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if length(X)/length(D) ~= obj.sps
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warning('Signal length does not fit to reference!');
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end
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% ==============================================================
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% INITIALIZATION (only before final epoch and detection mode)
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% ==============================================================
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% --- Parameters
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obj.S = numel(obj.constellation); % alphabet size
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obj.Nf = obj.order*obj.sps; % filter length
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% obj.delta = 3;%ceil(obj.Nf/2); % delay parameter
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obj.nStates = obj.S^obj.L;
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obj.nFeasible = obj.nStates*obj.S;
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% --- Trellis mapping
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obj.trellis_states = reshape(obj.constellation,1,[]);
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pre_comb_mat = repmat(obj.trellis_states, obj.L, 1);
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pre_comb_cell = mat2cell(pre_comb_mat, ones(1,obj.L), size(pre_comb_mat,2));
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obj.combs = fliplr(combvec(pre_comb_cell{:}).'); % rows: states, columns: [x_k, x_{k-1}, ...]
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obj.first_sym = obj.combs(:,1);
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obj.last_sym = obj.combs(:,end);
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obj.nStates = size(obj.combs,1);
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% --- Valid transitions
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obj.valid = false(obj.nStates);
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for from = 1:obj.nStates
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for to = 1:obj.nStates
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if all(obj.combs(to,2:end) == obj.combs(from,1:end-1))
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obj.valid(to,from) = true;
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end
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end
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end
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[obj.valid_to_idx, obj.valid_from_idx] = find(obj.valid);
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% --- Allocate vectors and weights
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% !! IF SHAPE FIT, then we already have smth there an we want
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% to start with the existing fitler-set
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if isempty(obj.w) || any(size(obj.w) ~= [obj.Nf+1,obj.nFeasible])
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obj.w = zeros(obj.Nf+1,obj.nFeasible); % filter weights per transition + bias tap
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obj.w = randn(obj.Nf+1,obj.nFeasible);
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end
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% --- Precompute dictionary for fast state lookup (sequence -> state)
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keys = cell(obj.nStates,1);
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for i = 1:obj.nStates
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keys{i} = obj.seq_key(obj.combs(i,:)); % combs row is already [x_k, x_{k-1}, ...]
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end
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obj.state_dict = containers.Map(keys, 1:obj.nStates);
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% ==============================================================
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% TRAINING
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% ==============================================================
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% Training Mode
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n = obj.len_tr;
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training = 1;
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obj.equalize(X.signal, D.signal,obj.mu_tr,obj.epochs_tr,n,training);
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obj.e_tr = obj.e;
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% ==============================================================
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% DD-Mode / Fixed Mode
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% ==============================================================
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% Decision Directed Mode
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n = X.length;
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training = 0;
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[y,y_vit]=obj.equalize(X.signal, D.signal,obj.mu_dd,obj.epochs_dd,n,training);
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X_viterbi = X;
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X.signal = y;
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X.fs = D.fs; %change sampling frequency of outgoing signal from fdac e.g. 2 sps to symbol spaced = fsym
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lbdesc = [num2str(obj.order),' tap FFE'];
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X = X.logbookentry(lbdesc); % append to logbook
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X_viterbi.signal = y_vit;
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X_viterbi.fs = D.fs; %change sampling frequency of outgoing signal from fdac e.g. 2 sps to symbol spaced = fsym
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lbdesc = [num2str(obj.order),'order FFE + PF + Viterbi'];
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X_viterbi = X_viterbi.logbookentry(lbdesc); % append to logbook
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end
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function [y,y_ref] = equalize(obj,x,d,mu,epochs,N,training)
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% ==============================================================
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% FFE + Whitening + ML-Based Branch Metric Estimation + Viterbi
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% ==============================================================
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debug = 1;
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showPlots = 1;
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% --- Input padding and preallocation
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y = zeros(N,1);
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% number of symbol steps in this block
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nSymbols = ceil(N/obj.sps);
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for epoch = 1:epochs
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% state metrics (log-domain costs): keep as column [nStates×1]
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pm = zeros(obj.nStates,1); % v_{k-1}(s′)
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c_hat = zeros(1,obj.nFeasible);
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v_tilde = zeros(1,obj.nFeasible);
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pred = zeros(nSymbols, obj.nStates, 'uint32');
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pm_sto = nan(obj.nStates, nSymbols,'like',pm);
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CE_accum = 0;
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%%% START IDX
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if training
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max_start = length(x) - ( (ceil(N/obj.sps)-1)*obj.sps + 1 );
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max_start = max(1, max_start); % safety
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start_sample = randi([1, max_start], 1); %rnd training; not really good
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start_sample = 1;
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end_sample = start_sample + (ceil(N/obj.sps)-1)*obj.sps;
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else
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start_sample = 1;%obj.len_tr;
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end_sample = N;
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end
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start_symbol = 1 + floor((start_sample - 1)/obj.sps); % ABSOLUTE symbol index
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if numel(d) >= obj.L && start_symbol >= obj.L
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init_seq = d(start_symbol-obj.L+1 : start_symbol); % [d_k-L+1 ... d_k]
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true_to_state_idx = obj.state_dict(obj.seq_key(flip(init_seq))); % [d_k ... d_k-L+1]
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else
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% Not enough history – fall back to state 1
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true_to_state_idx = uint32(1);
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end
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symbol = 0;
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for sample = start_sample:obj.sps:end_sample
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symbol = symbol + 1;
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k = symbol;
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sym_idx = start_symbol + (symbol - 1);
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% --- Build Δ-delayed observation window y_k
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i1 = sample - obj.Nf + 1 + obj.delta;
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i2 = sample + obj.delta;
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buf = x(max(1,i1):min(length(x),i2));
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padL = max(0,1 - i1);
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padR = max(0,i2 - length(x));
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yk = [zeros(padL,1); buf(:); zeros(padR,1)]; % Nf×1
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yk = [yk;1];
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% --- Predict branch metrics for all feasible transitions: c_hat
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c_hat = (yk.' * obj.w); % [1×nFeasible]
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c_hat = c_hat.'; % [nFeasible×1]
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% --- Extended path metrics: v_tilde = pm(from) + c_hat
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% normalize pm to avoid growth (invariant to additive const)
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pm = pm - min(pm);
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v_tilde = pm(obj.valid_from_idx) + c_hat; % [nFeasible×1]
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% ===== Gradient update (Algorithm 1) =====
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if 1 %training
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% --- allocate storage once
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if epoch == 1 && symbol == 1
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obj.true_to_state_idx = ones(ceil(N/obj.sps),1,'uint32');
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end
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% --- previous "to" becomes current "from"
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if symbol > 1
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true_from_state_idx = obj.true_to_state_idx(symbol-1);
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else
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true_from_state_idx = 1;
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end
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% --- compute or reuse "to" state
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if epoch == 1
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% only compute in first epoch
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if sym_idx >= obj.L
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key_to = obj.seq_key(flip(d(sym_idx-obj.L+1 : sym_idx)));
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if isKey(obj.state_dict, key_to)
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obj.true_to_state_idx(symbol) = obj.state_dict(key_to);
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else
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obj.true_to_state_idx(symbol) = true_from_state_idx;
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end
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else
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obj.true_to_state_idx(symbol) = true_from_state_idx;
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end
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end
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% --- reuse cached state from second epoch onward
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true_to_state_idx = obj.true_to_state_idx(symbol);
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% --- ensure valid (from,to)
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dirac = zeros(obj.nFeasible,1);
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mask = obj.valid_from_idx==true_from_state_idx & ...
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obj.valid_to_idx ==true_to_state_idx;
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if any(mask)
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dirac(mask) = 1;
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else
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idx = find(obj.valid_from_idx==true_from_state_idx,1,'first');
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dirac(idx) = 1;
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obj.true_to_state_idx(symbol) = obj.valid_to_idx(idx);
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end
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% softmax over -v_tilde (numerically safe shift)
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v_shift = -(v_tilde - min(v_tilde)); % shift to small positive numbers
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v_shift = min(v_shift, 100); % clamp exponent argument (≈ exp(50)=3e21)
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expv = exp(v_shift);
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p = expv ./ (sum(expv) + eps);
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% for logging only:
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CE_symbol(symbol) = -log(p(dirac==1) + eps);
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if sym_idx > obj.L
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CE_smooth(symbol) = 0.01*CE_symbol(symbol) + 0.99*CE_smooth(symbol-1);
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else
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if epoch > 1
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CE_smooth(symbol) = obj.ce(end); %use ce from last epoch or =1 for very first round?!
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else
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CE_smooth(symbol) = CE_symbol(symbol);
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end
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end
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CE_accum = CE_symbol(symbol) + CE_accum;
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% gradient term (t - p)
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dmp = (dirac - p)'; % 1×nFeasible
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% Per-feature gradient; implicit expansion gives (Nf+1)×nFeasible
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dL_Dw = (yk) .* dmp;
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% Start updates only when the ABSOLUTE symbol index has ≥ L history
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if sym_idx >= obj.L
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if obj.adaptive_mu
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mu_eff = CE_smooth(sym_idx);
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mu_eff = max(min(mu_eff, 0.2), 1e-4);
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else
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mu_eff = mu;
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end
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obj.w = obj.w - mu_eff .* dL_Dw; % (Nf+1)×nFeasible
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end
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% if debug && epoch > 2
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% figure(100);
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% subplot(4,1,1);
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% heatmap(p');
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% title('Probs')
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% subplot(4,1,2);
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% heatmap(dmp);
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% title('Update')
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% subplot(4,1,3);
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% heatmap(dL_Dw);
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% title('Update')
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% subplot(4,1,4);
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% heatmap(bj.w);
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% title('Update')
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%
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% end
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end
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% --- Compare-Select (matrix form, min of costs)
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v_tilde_mat = inf(obj.nStates, obj.nStates);
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v_tilde_mat(obj.valid) = v_tilde;
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[pm_next, pred(k,:)] = min(v_tilde_mat, [], 2);
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% re-center to keep metrics bounded (decision-invariant)
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pm_next = pm_next - min(pm_next);
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pm = pm_next;
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pm_sto(:,symbol) = pm;
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end
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% --- Traceback (full; you can window with traceback_depth if desired)
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[~, s_end] = min(pm);
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viterbi_path = zeros(symbol,1,'uint32');
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viterbi_path(symbol) = s_end;
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for n = symbol:-1:2
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viterbi_path(n-1) = pred(n, viterbi_path(n));
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end
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y_ref = d(start_symbol:end);
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y = obj.first_sym(viterbi_path);
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if debug && training
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sym_start = start_symbol;
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sym_end = start_symbol + symbol - 1;
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ref_slice = d(sym_start : sym_end);
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err = sum(y ~= ref_slice(1:numel(y)));
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try
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ref_bits = PAMmapper(obj.S,0).demap(ref_slice);
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eq_bits = PAMmapper(obj.S,0).demap(y);
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[~, ~, ber, ~] = calc_ber(ref_bits, eq_bits, "skip_front", 10, "skip_end", 10, "returnErrorLocation", 1);
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fprintf('Epoch: %d - BER: %.1e \n',epoch, ber);
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obj.ber(epoch) = ber;
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catch
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ser = err./length(y);
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fprintf('Epoch: %d - SER: %.1e \n',epoch, ser);
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end
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obj.ce(epoch) = CE_accum./symbol;
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if showPlots
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figure(10);clf
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subplot(3,2,1:2);
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heatmap(obj.w);
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title('Filter')
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subplot(3,2,3);
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v_tildemat = NaN(obj.nStates, obj.nStates);
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v_tildemat(obj.valid) = v_tilde; % log-domain scores
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heatmap(v_tildemat);
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title('Path Metrics (v_tilde)')
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subplot(3,2,4);
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scatter(1:symbol,pm_sto,1,'.')
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title('Path Metric Winners')
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subplot(3,2,5);hold on
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scatter(1:symbol,CE_symbol,1,'.');
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scatter(1:symbol,CE_smooth,1,'.')
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title('Cross Entropy')
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subplot(3,2,6); hold on
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% Left y-axis: Cross Entropy (linear)
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yyaxis left
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scatter(1:length(obj.ce), obj.ce, 10, 's', 'filled')
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ylabel('Cross Entropy')
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% Right y-axis: BER (logarithmic)
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yyaxis right
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scatter(1:length(obj.ber), obj.ber, 10, 'd', 'filled')
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set(gca, 'YScale', 'log')
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ylabel('BER (log scale)')
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xlim([1, epochs])
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xlabel('Epoch')
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title('Cross Entropy // BER')
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grid on
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drawnow
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end
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end
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end
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end
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end
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methods (Access=private)
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function k = seq_key(obj, seq)
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% Build a stable key string for a sequence row vector in the *same order as combs rows* ([x_k, x_{k-1}, ...])
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% Use rounding via sprintf to avoid floating-point issues.
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% seq must be a row vector.
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k = sprintf(obj.key_fmt, seq);
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end
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end
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end
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